Advanced Semiconductor Fundamentals Chapter 4 Equilibrium Carrier Statistics

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Chapter 4 Equilibrium Carrier Statistics Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics DENSITY OF STATES The density of state function per unit volume, g(E)d. E, gives the number of available quantum states in the energy interval between E and E + d. E. What do we need to know? i) E – k relationship (dispersion relation) If not : anisotropic effective mass ii) Proceed with constant energy surface. isotropic effective mass → spherical constant energy surface anisotropic effective mass → ellipsoidal constant energy surface Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics General Derivation d. E E + d. E E constant energy surface Total volume in k-space between E and E + d. E, constant energy surface Number of states/each unit volume = 1 = 2, including spin differential surface area for constant energy surface Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Total number of states between E and (E + d. E)lunit volume spin The differential energy, Then, general expression for density of states or equivalently, Then, volume element =2 heavy hole light hole split-off dimension = 1 for 1 -D 2 for 2 -D 3 for 3 -D Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Density of States for Free Particle in 3 - D parabolic E-k relationship spherical constant energy surface from Density of States for Free Particle in 2 - D and 1 - D i) 2 -D Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics ky area element dimension = 2 for 2 -D kx : constant ii) 1 -D kx g(E) E Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Specific Materials Conduction Band – Ga. As : spherical constant energy surface same as free particle except that Conduction Band – Si, Ge 6 equivalent band minima Let : ellipsoidal constant energy surface then, Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics : spherical constant energy surface in k’-space and Then, equivalent band minima Nel = 6 for Si (not at X-band edge) ∴ for Si for Ge Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Valence Band – Si, Ge, Ga. As Chapter 4. Equilibrium Carrier Statistics degenerated hh and lh band at k = 0 k heavy hole light hole split-off Neglecting split-off band, ∴ Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Density of States of Low-Dimensional Semiconductors i) 2 -D semiconductor (quantum well) Each subband has constant density states. E 4 E 3 E 2 E 1 ii) 1 -D semiconductor (quantum wire) E 1 E 2 E 3 Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics FERMI FUNCTION Energy distribution functions: Maxwell-Boltzmann : classical Bose-Einstein quantum mechanical Fermi-Dirac compose of the product of two terms: 1) the number of energy states with the energy interval 2) the probability that a particle occupies the states Fermi-Dirac distribution function is applied for the particle system which obeys the Pauli Exclusion Principle. No two electrons can have identical quantum states. Bose-Einstein distribution function is applied for the particle system which does not obey the Pauli Exclusion Principle. Fermi-Dirac Statistics Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Derivation Proper Each allowed state can accommodate one and only one electron. Two macroscopic constraints: : total number of particles constants : total energy of particle system What is the most probable distribution subject to these constraints? How many ways can we arrange Ni particles among the Si states in a given interval? Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Like putting Ni objects into Si boxes, but the boxes will hold only 1 particle. Level electrons we can put first in Si boxes, second into Si -1 boxes………last into Si – Ni + 1. Total number of different ways of putting labeled electrons into the boxes: . However, electrons are indistinguishable and there are Ni! Ways of labeling the electron. ∴ The number of physically different ways of putting Ni electrons among Si states in the Ei energy level is for any Ei level The total number of different ways in which N electrons can be arranged in the multilevel system (i. e. , number of ways putting N 1 electrons into S 1 states, N 2 electrons into S 2 states………Ni electrons onto Si states), Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Under thermal equilibrium, the most probable distribution or arranged of electrons is the one that is most disordered. That is, the distribution of electrons which can occur in the largest number of ways is the most probable one. ∴ The most probable distribution occurs for maximum W subject to the constant constraints of Maximize W with respect to Ni methods of “ Largrangian undetermined multipliers” Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Using Stirling’s approximation, If we define a Fermi energy, called “Fermi-Dirac distribution function” Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics We can define the Helmholtz function, and the Gibbs function as F and G are minimum at thermal equilibrium. Boltzmann definition of entropy at thermal equilibrium: : most probable arrangement of particles in a crystal. Substituting (b) into (a), Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics and Replacing Ei with continuous variable E in energy band, Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics SUPPLEMENTAL INFORMATION Equilibrium Distribution of Carriers Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics The Energy Band Diagram Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Donors, Acceptors, Band Gap Centers Intrinsic material, n = p = ni Extrinsic material Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Ionization(Binding) Energy of Donor and Acceptor Energy required for electron in solid to make a transition from the donor level to the conduction band become (quasi) free. r 3 n=3 e, m 0 hydrogen atom in vacuum r 1 e, m* r 2 n=2 = 11. 7 for Si n=1 r 1 donor atom in Si n=1 r 2 r 3 n=2 n=3 n = 3, E 3 n = 2, E 2 n = 1, E 1 hydrogen atom n=1, 2, 3, , Si n = , E = 0 n = 3, Ed 3 n = , Ed = Ec n = 2, Ed 2 n = 1, Ed 1 donor atom ~6 me. V E c= Ed n=1, Ed 1= Ed Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics EQUILIBRIUM CONCENTRATION RELATIONSHIPS Formulas for n and p Electron concentration in the conduction band Hole concentration in the valence band using change of variable, due to exponential dependence of f(E) then Similarly, for holes in valence band Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals where Chapter 4. Equilibrium Carrier Statistics : Fermi-Dirac integral of order 1/2 : effective density of states at conduction band edge : effective density of states at valence band edge Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics For nondegenerated semiconductors, If then likewise If Mass-action law Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Physical Meaning of “Effective Density of States” for nondegenerate Charge Neutrality Relationship From Poisson’s equation dopant sites totally ionized Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Relationships for ND+ and NADonor and Acceptor Statistics Gibbs grand sum chemical potential Probability in a state Fermi-Dirac distribution Occupancy Energy N 0 0 0 1 E 1 Probability of occupancy, with EF. Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Electrons in a band with spin Probability of occupancy Donors with EF. Probability of occupancy = probability that the donor atoms are unionized (neutral donors) where Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Probability of being unoccupied = probability that the donor atoms are ionized 2 where with EF. degeneracy factor for donors Acceptors hh lh Probability of occupancy = probability of being ionized acceptors where 4 Deep level trap centers with EF. degeneracy factor for acceptors for donor-like for acceptor-like where Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics CONCENTRATION AND EF CALCULATION Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Equilibrium Carrier Concentrations From charge neutrality and assuming nondegeneracy Freeze-out/extrinsic T (ND >> NA or NA >> ND ) In a donor-doped semiconductor (ND >> NA), or almost fully ionized at room temperature Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Extrinsic/Intrinsic T (relatively high T) Then, : charge neutrality : nondegenerated semiconductor For donor-doped semiconductor, extrinsic T (ND >> NA, ND >> ni), For acceptor-doped semiconductor, extrinsic T (NA >> ND, NA >> ni), For intrinsic T, For compensation, Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Determination of EF Exact position of Ei For intrinsic semiconductor, n = p, NA = ND = 0, EF = Ei Freeze-out/extrinsic T (ND >> NA or NA >> ND ) In a donor-doped semiconductor (ND >> NA, ND >> ni) or equivalently (at low temperature extrinsic region) if EF – ED > 0 and as T goes small. or Jung-Hee Lee @ Nitride Semiconductor Device Lab.

Advanced Semiconductor Fundamentals Chapter 4. Equilibrium Carrier Statistics Extrinsic/Intrinsic T (relatively high T) For ND >> NA and ND >> ni or For NA >> ND and NA >> ni or What happens for partially compensated donor and acceptor with ND > NA? Read “degenerate semiconductor consideration’ Jung-Hee Lee @ Nitride Semiconductor Device Lab.